Cutting a part into two objects using a plane
Partial answer: Don't select both the cube and the plane before going to the modifier.  Select the cube, specify the boolean modifier, specify difference, select plane, then apply.
This results in a part of the original cube (depending on the normals of the plane?) and the plane as the two surviving objects.  I would expect this result if I had started with a cube and another smaller cube and done a difference or an intersection. In other words, slicing with a plane (0 thickness) shouldn't result in loss of the original volume.